Variance of a Binomial
Name here
2022-09-28
Introduction
We have learnt that the MGF for a binimial random variable is:
\[M_X(t) = (q+pe^t)^n \]
Where \(X\sim Bin(n,p)\)
Tasks
Using the moment generating function above prove that
\[\sigma^2 = npq\]
by answering the tasks below. Use \(\LaTeX\) to construct the proof for Task 1.
Task 1
Start with
\[ \begin{eqnarray} E(X) &=& \left . \frac{d M_X(t)}{dt}\right |_{t=0}\\ &=& npe^t(q + pe^t)^{n - 1}\\ \end{eqnarray} \]
Task 2
You may use paper and write neatly the proof - take a picture and place in the document using
![](){}
Now find \(E(X^2)\)
\[
\begin{eqnarray}
E(X^2) &=& \left . \frac{d^2 M_X(t)}{d^2t}\right |_{t=0}\\
&=&
\end{eqnarray}
\] 
Task 3
Find
\(\sigma^2\) Using the formula \(\sigma^2 =E(X^2)-\mu^2\) 